


Blood, Sex, and Lebesgue Integrals

by Bitenomnom



Series: Mathematical Proof [17]
Category: Sherlock (TV)
Genre: (okay - fake blood), Blood, John is the World's Most Tolerant Flatmate, M/M, Mathematics, No actual porn, Sherlock makes a pun, Sherlock uses John as a crime scene, with sexy results
Language: English
Status: Completed
Published: 2012-10-03
Updated: 2012-10-03
Packaged: 2017-11-15 13:11:44
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,612
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/527683
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>Blood, chocolate syrup, John in his pants on the floor, and a stupidly simple answer that was only stumbled upon because of sex.<br/>Not a bad day for John, considering he got kidnapped (again) at lunch.</p>
            </blockquote>





	Blood, Sex, and Lebesgue Integrals

**Author's Note:**

> I was so dreading working on this because I had absolutely no ideas for what to do. (Also this class is kicking my butt. Like, I almost cried at my homework grade today. In light of that, please correct me if I have made any errors on the math bit here.) I don't know what magical land the idea for this one came from, and I don't want to curse it by asking. And sorry for the lack of actual porn, again. I feel like such a tease but it's just that I (unlike Sherlock) occasionally require sleep and slapdash erotica sounds like a bad idea. (Ahahaha. Or a very good one, if it involves actual slapping and some brief sprinting, I suppose.)
> 
> A friend and I used a similar recipe to that mentioned in this story for a silly little zombie movie we made a few years back.

Lebesgue integrals can be used to integrate a wider variety of functions than the usual Riemann integrals can. (Every function that is Riemann integrable is also Lebesgue integrable.) For instance, take the function

f(x) =   0 if x is rational

            1 otherwise

There are countably infinite rational numbers here (even on some range, say, [0,1]), and uncountably infinite irrational numbers. This means that no matter how small an interval we choose (such as in Riemann integrals, when our partitions approach width 0), we can always find rational and irrational numbers in the range. So the upper sum of the Riemann integral will always be 1, and the lower sum will always be 0. Therefore, while we might intuitively expect the integral of this function over [0,1] to equal 1, since there are more irrational than rational numbers, we cannot perform this integration.

We can, however, perform it using the Lebesgue integral. In this approach, you divide horizontally and check how many points there are such that the value of f is in that interval (measure of the set of the points in the range). These points form a measurable set. Therefore, in an example like the previous one, in the top interval of the function, with the irrational numbers, you measure the set and multiply that by the height (1). For the rational numbers, the height is 0, and therefore the function is integrable with value 1.

In effect, you are sort of turning a “usual” integral on its side.

***  
  
            “John, I need you to do something for me.”

            “Did it ever occur to you that I might occasionally be busy doing my own sodding thing, Sherlock?”

            “Like what? What could be more important than this?”

            John buried his head in his hands. “Trying to hold onto the last scraps of my sanity, for a start.”

            “Boring,” Sherlock said, and then, “Mycroft had something to tell you today, didn’t he?”

            “If by that you mean, ‘Did my unbearable git of a brother kidnap you again today, John?’ the answer is yes.”

            Sherlock smirked. “He _is_ an unbearable git, isn’t he? What was it this time—an attractive woman slightly too interested in you? Or the usual intimidating ‘you-know-what-this-means’ black government automobile?”

            “A patient in the clinic, Sherlock. _A patient_ so politely told me that if I knew what was best for me I’d take my lunch break straight away and go with him.”

            “And what did he have to say?—Wait, don’t answer that. He wants you to try to convince me to investigate a particular case, which just so happens to be of interest of him?” John sighed and glanced to the side—a yes, then. “He’s given up on Lestrade asking, so either it’s urgent or he knows he’s been pushing it the past few months. _Or_ ,” his eyes lit up, and then narrowed. “No…no, certainly Lestrade must have better judgment than that.”

            “Okay,” John said, not even the slightest bit interested in what Sherlock was about to go on about primarily due to a marked disinterest in even the briefest of mentions of Mycroft at just this moment. “I’m tired of talking about this. Let’s agree that I told you about the case and you’re uninterested because you’re working on something else.”

            “I _am_ work on something else. I just requested your assistance. That’s what you do, isn’t it? Assist me?”

            “I’d like to think I’m a doctor,” John suggested.

            “Right,” Sherlock waved his hand limply, dismissing the idea. “Yes, but, _really_ , you are my assistant.”

            John opened his mouth to argue but then realized he had no particularly effective counter to that argument. It had gotten to that point—his locum work was more or less an excuse to leave the flat during their downtime, so that he didn’t have to listen to Sherlock’s occasionally unbearable violin-plucking and could spend a sliver of his day not monitoring his surroundings for explosions, noxious fumes, or rogue fingers. In the end and when he had to choose, it was always Sherlock.

            “All right. What is it?”  
            “The marks of blood flow across the body at the crime scene,” Sherlock nodded toward a cold case file Lestrade had given him earlier that day, “make absolutely no sense.”

            “You want me to have a look?”

            “If you’d like, but that wasn’t what I was going to request,” Sherlock handed him the photos, but clearly didn’t expect John to glean more from it than he had.

            “All right,” John said, flipping through the photographs. “That is odd, I agree. What was your idea?”

            Sherlock pulled out a jar full of… “Mostly peanut butter, corn syrup, and chocolate syrup. Food coloring added for effect and enhanced observation of trail of flow,” he smirked to himself. Of course, what it looked like was blood—fake blood, yes, certainly, John thought, but blood. And, of course, there went John’s plans for ice cream tonight.

            “I believe the orientation of the victim as or after the bleeding occurred is key to the irregular patterns.”

            “Right, so shall I get out some paper or something?”

            “No, John.” Sherlock stirred the stuff with a spoon he’d left in the container. “I need to test it on a body.”

            “You’re kidding.”

            “What about my suggestion don’t you understand? The blood flowed over a body. Paper will yield completely inaccurate results.”

            “I am not letting you put syrup all over my clothes.”

            “Better take them off, then. And hurry, if you don’t mind, before the consistency of our simulated blood is completely off.”

            John massaged his temples. There was no one else he could suggest to do this instead. He wouldn’t inflict this upon anyone. Sherlock had a fair point about using a human body—paper wasn’t the right shape, and the mixture would probably bleed through it. There weren’t any appropriately shaped pieces of furniture in the flat he’d be willing to sacrifice, short of perhaps putting plastic wrap around some of the pillows. “I’m keeping my pants on,” he insisted. “And don’t you even think about covering them in chocolate syrup.”

            “It’s not _just_ chocolate syrup, John.”

            “That’s not my point and you know it.”

            Sherlock shrugged. “Due to the nature of the experiment I cannot promise any clothing left upon your person will remain unperturbed.”

            John pulled his jumper over his head and tossed it onto the sofa. “I can’t believe I’m doing this,” he groaned, trousers and undershirt following. He yanked his socks off. “We’re going to put a tarp over the carpet,” he insisted. “Mrs. Hudson will kill us if she sees this stuff all over the floor.”

            “Of course.” Sherlock seemed to be prepared, and was already unfolding it. “Now, down you go. Lie on your back, if you please.”

            “You owe me new chocolate syrup,” John said, as if clinging onto this would allow him to preserve the last of his dignity as he laid on the tarp in nothing but his pants, hoping and praying Sherlock could avoid making any cutting observations about his shoulder or his belly or whatever else might serve as “interesting” to Sherlock.

            “I foresaw such complications and nipped that problem in the bud.”

            “You bought more chocolate syrup?”

            “This may be cold,” Sherlock ignored him and dipped the spoon into the mixture, ladling it out over John’s chest.

            John shuddered under the first touch of the stuff to his skin—it _was_ cold. “I always thought this would happen to me under much less platonic circumstances,” John observed, staring at the ceiling.

            Sherlock paused midway through spooning up more syrup. “What?” He tilted his head. “How are solving crimes and sexual intercourse related?”

            “I meant being covered in chocolate syrup.”

            “It’s not _just_ chocolate s—” Sherlock paused, eyes darting over John, himself, the fake blood. John swore he saw Sherlock’s ears grow pink. “Oh. You meant—well, I suppose an addendum _could_ be arranged relating to cleanup of syrupy substances from the human body, but—”

            “I wasn’t suggesting—I mean—not that— _us—_ ” John sputtered. “I meant—in general. You know. A general person. Uh. Woman. Putting chocolate on—my—body.”

            “Oh,” Sherlock resumed ladling. “Yes. Of course.” John tried not to wince as the stuff dribbled onto him, some of it running down the sides of his ribcage and tickling his back. Sherlock glanced at one of the photographs and held the next spoonful over John’s face, carefully dribbling it on the side of his mouth, the cold metal of the spoon resting between John’s lips. He fought the urge to poke his tongue out and lick it, to see what the stuff tasted like. He fought, and he lost, and he swiped a lick of the stuff. It wasn’t bad, really. “John!” Sherlock huffed.

            “Sorry.”

            “Don’t talk,” Sherlock said. “You’re ruining the pattern.”

            John rolled his eyes.

            “So what is the purpose,” Sherlock wondered aloud, “of covering oneself in chocolate syrup in a sexual act? Why not just consume the syrup directly from its container?” As if he could sense John beginning to open his mouth, Sherlock held up a finger and spooned up more syrup, this time letting it slide down John’s collarbone. He tried not to squirm. “Hardly the most efficient means of calorie intake, and certainly chocolate syrup on ice cream would be widely considered more pleasant to eat overall than chocolate syrup on its own with the slight aftertaste of flesh.” Sherlock frowned at the idea. “Skin can’t possibly be that flavorful. Shh,” he said, as John struggled to keep himself from speaking again. “Legs now,” Sherlock explained as he moved down the tarp. John refused to watch Sherlock’s eyes as they traveled down his body, for fear of what parts of him they might linger on—not that it meant anything, and not that Sherlock hadn’t seen him wander out in a towel a time or two, but even running about London on a more or less weekly basis John wasn’t as fit as he used to be. Sherlock, on the other hand—Sherlock was—well. Like some kind of a bloody statue, with all those toned muscles and all that smooth skin. John shut his eyes. _Bad timing,_ he thought, and willed himself to focus on how uncomfortable the fake blood was becoming as it ‘coagulated’ and grew sticky on his chest and face.

            “There’s also the question over whether such acts are meant to be applied only to erogenous zones, or to the entire body,” Sherlock continued to ponder. “In theory, and again for maximum efficiency, one would assume the former; however, as many a poorly written sex guide has attempted to state, an indirect approach is allegedly superior—teasing, as they say.” John hoped the stuff would wash off easily as he felt gobs of it settle onto the hairs on his legs. Thinking about that was, of course, a great deal easier than paying too much attention to Sherlock and therefore making a concerted effort not to reply to his musings.

            “After all, for some persons stimulation in one of the less obvious areas is equally effective as, say, direct genital contact.” He paused, thoughtfully, and John wondered if they were almost done. Sherlock had been considerate enough to avoid dribbling fake blood on his pants, but aside from one arm and his feet, all of his front side was decorated in some amount of fake, syrupy blood. “For instance,” Sherlock continued quietly, and leaned down, perhaps to inspect his handiwork—John could hardly tell without craning his neck, which Sherlock would probably argue required a complete redo of this trial. John felt breath against his legs, and then his feet. “Your toes are wiggling.”

            Oh.

            So they were.

            John consciously stilled them.

            A few seconds later, he felt cold, thick liquid running down his toes, chilling lines from the top of his foot to his ankles and along the soles. His toes curled when one drop found its way to the arch of his foot. “Sherlock,” he said, and Sherlock didn’t shush him, “there wasn’t blood on the victim’s feet.”

            “Supplementary data,” was all Sherlock said, his voice quiet and wistful as it got when he fixated on something and retreated back into his own mind to process it away from the eyes of any onlookers—silently, unlike so many of his deductions.

            “You know, this isn’t exactly comfortable,” John reminded Sherlock as he sat there, thinking. “Can we get to the next step?”

            Sherlock nodded distantly and reached up to John’s chest, to some of the first syrup he had put down, and ran his finger through it. John shivered. “Consistency’s all wrong,” he muttered.

            Oh no. “Tell me I didn’t just hear that.”

            “I wasn’t expecting it to become so thick so quickly. I thought body heat would be enough to—”

            “Sherlock Holmes, do _not_ tell me you just made me strip and get myself doused in chocolate syrup for no reason whatsoever, and will have to do so again.”

            “We should make the best of the current situation,” Sherlock said, voice still distant.

            “Oh, great idea, let’s—” _oh._ Oh. _Oh._

            His toes were suddenly hot. John was reasonably certain that no high-pitched sighs escaped his throat in response to the sensation. He most particularly did not squirm or wiggle his hips at the twisting of a deft tongue around and between his toes. He probably completely mussed up the fake blood on his arms when he reached out to grip the tarp as his not-at-all-wiggling hips lifted slightly off the ground while he groaned.

            Then Sherlock was up by his face again, staring at his eyes with the sort of attention generally reserved for dead bodies. “Not good?”

            “No,” John said, and sod it, sod it all, he was already almost undressed and had just been covered in chocolate syrup by his flatmate and after being kidnapped by the British Government at lunch that constituted a pretty good day, and there was no reason to go spoiling what anyone in the world would use as evidence they were a couple by insisting otherwise just out of habit. Maybe they were right. John couldn’t give a bloody fuck just now. Or, he thought, maybe he could.

 

 

 

            “Good job we put the tarp down,” Sherlock said, lying beside John on the now even stickier surface. “We’ll have to clean it before we try again, of course.”

            “I was thinking we could try a bed next time,” John suggested wryly.

            “I was talking about the blood experiment,” Sherlock said, and then realized John had been joking, and let loose one of his genuine, sunny smiles. “On a related note, however, increased ambient heat is helpful in keeping the fake blood at an appropriate viscosity.” He nodded toward John, who was still covered in more chocolate-syrup-mixture than he generally preferred to be.

            “Yeah,” John flipped to his side to prop his head up on his hand, “well, in that case, let’s do that _before_ putting blood on me next time, shall we?”

            “Agreed,” Sherlock nodded, and then his eyes widened. John followed the direction of Sherlock’s gaze to his stomach. Well, he was bound to say— “That’s it.”

            John blinked.

            “Shortly after the victim was attacked and subsequently murdered, he was turned on his _side_ ,” Sherlock said. “But only for a time. Prior to that, the blood had coagulated slightly—only very slightly—but on his side, he had been facing toward the fireplace.”

            “And there’s your answer.”

            “Yes.”

            “No, I mean to your other question.”

            “What other question?”

            “‘How are solving crimes and sexual intercourse related?’”

            “Ah!” Sherlock’s face lit up. “Yes. _Intimately_.”


End file.
